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# Source file src/math/atan.go

## Documentation: math

```     1  // Copyright 2009 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
4
5  package math
6
7  /*
8  	Floating-point arctangent.
9  */
10
11  // The original C code, the long comment, and the constants below were
12  // from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
13  // http://www.netlib.org/cephes/cmath.tgz.
14  // The go code is a version of the original C.
15  //
16  // atan.c
17  // Inverse circular tangent (arctangent)
18  //
19  // SYNOPSIS:
20  // double x, y, atan();
21  // y = atan( x );
22  //
23  // DESCRIPTION:
24  // Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
25  //
26  // Range reduction is from three intervals into the interval from zero to 0.66.
27  // The approximant uses a rational function of degree 4/5 of the form
28  // x + x**3 P(x)/Q(x).
29  //
30  // ACCURACY:
31  //                      Relative error:
32  // arithmetic   domain    # trials  peak     rms
33  //    DEC       -10, 10   50000     2.4e-17  8.3e-18
34  //    IEEE      -10, 10   10^6      1.8e-16  5.0e-17
35  //
36  // Cephes Math Library Release 2.8:  June, 2000
37  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
38  //
39  // The readme file at http://netlib.sandia.gov/cephes/ says:
40  //    Some software in this archive may be from the book _Methods and
41  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
42  // International, 1989) or from the Cephes Mathematical Library, a
43  // commercial product. In either event, it is copyrighted by the author.
44  // What you see here may be used freely but it comes with no support or
45  // guarantee.
46  //
47  //   The two known misprints in the book are repaired here in the
48  // source listings for the gamma function and the incomplete beta
49  // integral.
50  //
51  //   Stephen L. Moshier
52  //   moshier@na-net.ornl.gov
53
54  // xatan evaluates a series valid in the range [0, 0.66].
55  func xatan(x float64) float64 {
56  	const (
57  		P0 = -8.750608600031904122785e-01
58  		P1 = -1.615753718733365076637e+01
59  		P2 = -7.500855792314704667340e+01
60  		P3 = -1.228866684490136173410e+02
61  		P4 = -6.485021904942025371773e+01
62  		Q0 = +2.485846490142306297962e+01
63  		Q1 = +1.650270098316988542046e+02
64  		Q2 = +4.328810604912902668951e+02
65  		Q3 = +4.853903996359136964868e+02
66  		Q4 = +1.945506571482613964425e+02
67  	)
68  	z := x * x
69  	z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
70  	z = x*z + x
71  	return z
72  }
73
74  // satan reduces its argument (known to be positive)
75  // to the range [0, 0.66] and calls xatan.
76  func satan(x float64) float64 {
77  	const (
78  		Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
79  		Tan3pio8 = 2.41421356237309504880      // tan(3*pi/8)
80  	)
81  	if x <= 0.66 {
82  		return xatan(x)
83  	}
84  	if x > Tan3pio8 {
85  		return Pi/2 - xatan(1/x) + Morebits
86  	}
87  	return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
88  }
89
90  // Atan returns the arctangent, in radians, of x.
91  //
92  // Special cases are:
93  //      Atan(±0) = ±0
94  //      Atan(±Inf) = ±Pi/2
95  func Atan(x float64) float64
96
97  func atan(x float64) float64 {
98  	if x == 0 {
99  		return x
100  	}
101  	if x > 0 {
102  		return satan(x)
103  	}
104  	return -satan(-x)
105  }
106
```

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